Abstract
In this paper, quadrature formulas on the unit circle are considered. Algebraic properties are given and results concerning error and convergence established.
Finally, numerical experiments are carried out.
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References
W. Gautschi,A survey of Gauss-Christoffel quadrature formulae, in E.B. Christoffel; The Influence of his Work in Mathematics and the Physical Sciences, eds. P.L. Butzer and F. Fehér, Birkhäuser, Basel (1981) pp. 72–147.
W.B. Jones, O. Njåstad and W.J. Thron,Moment theory, orthogonal polynomials, quadrature and continued fractions associated with the unit circle, Bull. London Math. Soc. 21 (1989) 113–152.
A. Bultheel, P. González-Vera, E. Hendriksen and O. Njåstad,Orthogonality and quadrature on the unit circle, IMACS Annals on Computing and Applied Mathematics, vol. 9. (1991) pp. 205–210.
V.I. Krylov,Approximate calculation of integrals (Macmillan, 1962) (translated by A.H. Stroud).
W. Gautschi and G.V. Milovanović,Polynomials orthogonal on the semicircle, J. of Approximation Theory, vol. 46 (3) (1986) 230–250.
W. Gautschi, H.J. Landau and G.V. Milovanović,Polynomials othogonal on the semicircle, II, Constructive Approximation 3 (1987) pp. 389–404.
P.J. Davis and P. Rabinowitz,Methods of Numerical Integration, 2nd ed. (Academic Press, 1984).
M. Camacho and P. González-Vera,A note on para-orthogonality and biorthogonality, Det Kongelige Norske Videnskabers Selskab Skrifter 3(1992) pp. 1–16.
H. Waadeland,A Szegö quadrature formula for the Poisson integral, Comp. and Appl. Math. I, eds. C. Brezinski and U. Kulish (Elsevier Science Publishers B.V., North-Holland, 1992) 479–486.
K.E. Atkinson,An introduction to numerical analysis, 2nd ed. (Wiley, New York, 1989).
C. Brezinski,Padé-type approximation and general orthogonal polynomials (Birkhäuser Verlag, 1986).
S. Wolfram,Mathematica: A system for doing Mathematics by computer, 2nd ed. (Addison-Wesley, 1991).
P. González-Vera,On certain applications of two-point Padé-type approximants, J. of Computational and Appl. Maths. vol 19 (1) (1987) pp. 151–160.
N.I. Akhiezer,The classical moment problem and some related questions in analysis (Hafner, New York, 1964) (translated by N. Kemmer).
G. Szegö,Orthogonal polynomials, American Mathematical Society, 4th ed., vol. 23 (Colloquium Publications, 1978).
W.B. Gragg,Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle, J. of Comput. and Appl. Math. 46 (1993) 183–198.
T. Bloom, D.S. Lubinsky and H. Stahl,Interpolatory integration rules and orthogonal polynomials with varying weights, Numerical Algorithms 3 (1992) 55–66.
A. Bultheel, P. González-Vera, O. Njåstad and E. Hendriksen,On convergence of multipoint Padé Approximants associated with the unit circle. Submitted.
A. Bultheel, P. González-Vera, O. Njåstad and E. Hendriksen,Quadrature formulas on the unit circle and two point Padé approximation, Nonlinear numerical methods and rational approximation II, ed. A. Cuyt (Kluwer, 1994) 303–318.
E. Godoy and F. Marcellán,Orthogonal polynomials and rational modifications of measures, Can. J. Math. vol 45 (5) (1993) pp. 930–943.
P. González-Vera, R. Orive and J.C. Santos-León,Quadrature formulas with nodes on the unit circle, Actas III Congreso de Matematica Applicada, eds. A. Casal et al. (1995) 609–615.
P.J. Davis,On the integration of periodic analytic functions, On Numerical Approximation, ed. R.E. Langer (Univ. Wisconsin Press, Madison, 1950) pp. 45–59.
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Communicated by C. Brezinski
This research was performed as part of the European project ROLLS under contract CHRX-CT93-0416.
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González-Vera, P., Santos-León, J.C. & Njåstad, O. Some results about numerical quadrature on the unit circle. Adv Comput Math 5, 297–328 (1996). https://doi.org/10.1007/BF02124749
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DOI: https://doi.org/10.1007/BF02124749